I live in Kansas, where grass is one tough customer! Depending on the variety of grass, it really will grow "as tall as the withers on a horse", as the saying went.
The think is, wheat is also grass, and it also does well here.
From what I have seen and done in my own back
yard, you might try mowing the grass in the Fall and then planting winter wheat in the now-short grass. When I speak of planting, I mean to broadcast the seed over the top of the mown grass just before a rain. I have some
volunteer wheat in my lawn, and I sprinkled some seed on the lawn in the spring. The volunteer wheat from last Fall did the best by far.
I am leaving some of it for another month, at which time it
should be ripe. I can get seed off of it and re-sow it in a less VISIBLE area!!!